Problem: Complete the square to solve for $x$. $x^{2}+x-30 = 0$
Move the constant term to the right side of the equation. $x^2 +x = 30$ We complete the square by taking half of the coefficient of our $x$ term, squaring it, and adding it to both sides of the equation. The coefficient of our $x$ term is $1$ , so half of it would be $\dfrac{1}{2}$ , and squaring it gives us ${\dfrac{1}{4}}$ $x^2 +x { + \dfrac{1}{4}} = 30 { + \dfrac{1}{4}}$ We can now rewrite the left side of the equation as a squared term. $( x + \dfrac{1}{2} )^2 = \dfrac{121}{4}$ Take the square root of both sides. $x + \dfrac{1}{2} = \pm\dfrac{11}{2}$ Isolate $x$ to find the solution(s). $x = -\dfrac{1}{2}\pm\dfrac{11}{2}$ The solutions are: $x = 5 \text{ or } x = -6$ We already found the completed square: $( x + \dfrac{1}{2} )^2 = \dfrac{121}{4}$